Music And Measure Theory

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Recommends flawr · 2015-11 · Mathematics Stack Exchange · on “Mathematical YouTube channels?”

I recommend 3Blue1Brown's Channel [LINK] . For an example, see this video [this video] . Also The Mathologer [LINK] EDIT: I'd also like to add a newer channel PBS Infinite Series [LINK] (EDIT2: Unfortunately they already stopped making videos.)

Recommends M. Winter · 2017-09 · Mathematics Stack Exchange · on “Apparent inconsistency of Lebesgue measure”

This strange set you constructed is discussed in this video [this video] . It is known for its paradoxical property of being of finite measure, but having all rationals as *inner* points. Still, it is not all of $\Bbb R$.

Recommends ex.nihil · 2019-01 · Mathematics Stack Exchange · on “Expressing musical patterns as a mathematical model?”

There are too many aspects of music that you can talk about: tonality, dynamics, nuance, timbre, etc. It is not so simple to map all of them simultaneously. You have to narrow it down. There are some items from physics, signal processing, and acoustics: harmonics (ratios of intervals as they relate to signals). 3Blue1Brown , a YouTube creator with a strong math background, very elegantly presents this here: Music and Measure Theory [this video] Hope this helps, and if you need more ideas, I have enough background in both music and math to be helpful.

Recommends user3733558 · 2021-02 · Mathematics Stack Exchange · on “Showing how much bigger $|\mathbb{R}|$ is than $|\mathbb{N}|$”

In your voice: "Let's consider all the fractions of the form $1/n$. They are just as numerous as $\mathbb{N}$, right? How dense do you think they are in the $[0,1]$ interval? Wait, scratch that, let's go crazy and consider all fractions of the form $n/d$ with $n<d$." And then you argue like 3Blue1Brown did in [this video] (starting at 5:30, but watch it all if you haven't already)

Recommends vaebnkehn · 2021-03 · Mathematics Stack Exchange · on “Measuring Irrationality à la 3Blue1Brown vs Liouville”

So, I was watching this 3B1B video on measure theory and music [this video] (as you do) and was struck by how Grant uses epsilon to measure "how irrational" a real number is. He takes all the rationals $[0, 1]$ and assigns them progressively smaller ranges to cover as follows: Where each range is centered on its corresponding rational and has a width corresponding to the nth number in the sequence shown above (where $\epsilon \in [0, 1]$ obviously). He then claims that these ranges won't cover $\frac {\sqrt 2}{2}$ when $\epsilon = 0.3$ So, my main question is: how did he determine that? Does $\frac {\sqrt2}{2}$ have a continuing fraction representation which shows it "hiding" in the gaps between

Cites More Anonymous · 2019-11 · Mathematics Stack Exchange · on “What is the measure of $\int_{A}^B a_{\frac{x-A}{dx} } f(x) dx$?”

Mathematicians like to ask covering various sets with open intervals and the answers to these riddles have strange tendencies to become strange lemma's or theorems Heine-Borel Theorem, lebesgue's Number Lemma, Vitali Covering Lemma, Besicovitch Covering Theorem, (etc) - 3blue1brown [this video] By open interval he means the stretch of real numbers: $a<x<b$ . My challenge is to strictly weighted-ly cover all the rational numbers between $A$ and $B$ with open intervals. By weighted-ly cover means each paticular rational number lies in a particular interval. However, each interval has a weight $a_i$ . The challenge here is that weighted sum must be less than $k'$ . Before we proceed let's return to

Cites Adam Gluntz · 2025-04 · Mathematics Stack Exchange · on “How does this measure theory proof not result in a paradox?”

As a starter, it may be the case that this question goes beyond my math background, which peters out at linear algebra. However, my intuition tells me that this is either a paradox or perilous close to one if we aren't careful. This is an argument that I saw in 3blue1brown on YouTube (right here [this video] ) that shows that it is possible to cover the interval between 0 and 1 with open intervals such that all rational numbers lie inside at least one interval and where the sum of the length of all of those intervals is strictly less than 1. The argument is as follows: Let $S = \{x|(x\in\mathbb{Z}) \cap (0<x<1) \}$ and put all of the elements of $S$ onto an infinite list (such as you would when p